SHSAT Arithmetic prep
No calculators. Every algebra and geometry problem on the SHSAT requires accurate arithmetic underneath. Here's what's tested and how to build the calculator-free fluency the test rewards.
- What arithmetic is on the SHSAT?
- Integer operations, fractions, decimals, percents, ratios and proportions, basic number theory (primes, factors, GCF, LCM), and order of operations. Roughly 15–20% of math questions, concentrated in the easier portion of the section.
- Do I need to memorize anything?
- Multiplication tables to 12×12, common fraction-decimal-percent equivalents (1/4 = 0.25 = 25%, 1/3 ≈ 0.333, 1/8 = 0.125, etc.), prime numbers up to 100, and squares up to 15². These should be automatic — slow recall here costs time on every question.
- Why is arithmetic still important if I'm strong in algebra?
- The SHSAT doesn't allow calculators. Every algebra problem, every geometry problem, every word problem requires accurate hand-arithmetic. A student strong in algebra but slow at multiplication will make pacing errors throughout the section.
- What's the most common arithmetic error?
- Sign errors with negative numbers, followed by mistakes with negative exponents and order-of-operations errors involving parentheses inside fractions.
The arithmetic topics that matter
Integer operations
Addition, subtraction, multiplication, and division of positive and negative integers. Most SHSAT students have this mastered, but watch for:
- Sign rules with multiplication and division: negative × negative = positive, negative × positive = negative. Trivial when isolated, but easy to lose track of in multi-step problems.
- Subtraction of negatives: 7 − (−3) = 10. Common to misread as 7 − 3 = 4.
- Order of operations (PEMDAS): Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right). The SHSAT specifically tests cases where misapplied PEMDAS gives a tempting-but-wrong answer.
Fractions
The arithmetic topic where most students lose easy points. Key skills:
- Adding/subtracting: Common denominator required. 1/3 + 1/4 = 4/12 + 3/12 = 7/12.
- Multiplying: Multiply numerators, multiply denominators. (2/3) × (3/4) = 6/12 = 1/2. Look for opportunities to cancel before multiplying — the cancellation makes the arithmetic easier.
- Dividing: Multiply by the reciprocal. (2/3) ÷ (4/5) = (2/3) × (5/4) = 10/12 = 5/6.
- Mixed numbers and improper fractions: Convert between freely. 2 3/4 = 11/4 when you need to multiply or divide; keep it as 2 3/4 when comparing to other mixed numbers.
- Comparing fractions: Cross-multiply to compare. To compare 3/8 vs 5/12: 3 × 12 = 36 vs 5 × 8 = 40, so 5/12 is larger.
Decimals
- Multiplication: Multiply as if integers, then count decimal places. 0.4 × 0.3: multiply 4 × 3 = 12, then move two places left = 0.12.
- Division: Shift decimal in both numerator and denominator until denominator is whole. 12.5 ÷ 0.25 → 1250 ÷ 25 = 50.
- Decimal-fraction conversion: 0.6 = 6/10 = 3/5. Be fluent in both directions.
Percents
"Percent" means "per 100." Treat percent problems as either:
- Translation: "20% of 50" = 0.20 × 50 = 10. "What percent of 80 is 20?" = 20/80 = 0.25 = 25%.
- Percent change: (new − old) / old × 100. Going from 50 to 65 is (65−50)/50 = 0.30 = 30% increase.
- Successive percents: A 20% increase followed by a 20% decrease does NOT return to the original. 100 → 120 → 96. The percents apply to different bases.
- Sale price: A 25% discount on a $80 item = $80 × 0.75 = $60. Multiply by (1 − discount), not by the discount itself.
Ratios and proportions
A ratio compares quantities (3:4 or 3/4). A proportion sets two ratios equal (a/b = c/d).
- Solving proportions: Cross-multiply. 3/4 = x/20 → 4x = 60 → x = 15.
- Combined ratios: If a:b = 2:3 and b:c = 4:5, find a:b:c. Scale to make b consistent: a:b = 8:12, b:c = 12:15, so a:b:c = 8:12:15.
- Ratio as fraction of whole: If boys:girls = 3:5 in a class of 32, boys = (3/8) × 32 = 12.
Number theory
- Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Memorize these.
- Prime factorization: 60 = 2² × 3 × 5. Essential for GCF and LCM problems.
- GCF (Greatest Common Factor): Product of shared prime factors with the lowest power. GCF(12, 18) = 6 because 12 = 2²×3, 18 = 2×3², shared = 2×3 = 6.
- LCM (Least Common Multiple): Product of all prime factors with the highest power. LCM(12, 18) = 36.
- Divisibility rules: by 2 (last digit even), by 3 (digit sum divisible by 3), by 4 (last two digits divisible by 4), by 5 (last digit 0 or 5), by 9 (digit sum divisible by 9), by 10 (last digit 0).
How to study arithmetic for the SHSAT
- Build speed before accuracy. Untimed accuracy is necessary but not sufficient. The SHSAT rewards students who can do arithmetic accurately and quickly.
- Practice without a calculator. Always. Even when learning a new concept, work through examples by hand to build the arithmetic stamina the test requires.
- Drill fraction operations especially. Most students underestimate how much fraction speed affects their math section pacing. Spend 15 minutes per study session on pure fraction drills early in prep.
- Memorize the decimal-fraction-percent equivalents. 1/2 = 0.5 = 50%, 1/3 ≈ 0.333, 1/4 = 0.25 = 25%, 1/5 = 0.2 = 20%, 1/8 = 0.125, 3/8 = 0.375, 5/8 = 0.625, 7/8 = 0.875, 1/6 ≈ 0.167, 5/6 ≈ 0.833. These appear constantly in disguised forms.
- Watch for "easy" questions you might rush. Arithmetic questions tend to be earlier in the section. Resist the temptation to fly through them — the accuracy cost of careless errors is higher than the time cost of working carefully.
Common questions
Are calculators allowed on the SHSAT?
No. The SHSAT does not allow calculators on either the math or ELA section. All arithmetic must be done by hand. Practice without a calculator from the start of prep to build the arithmetic speed and accuracy the test requires.
How many arithmetic questions are on the SHSAT?
Approximately 15–20% of math questions test arithmetic concepts directly (fractions, decimals, percents, ratios, number theory). Many more questions test other topics but require arithmetic skill to solve — so functionally, arithmetic fluency affects performance on most of the math section.
Should I memorize prime numbers?
Yes, at least through 50. Knowing primes lets you do prime factorization quickly, which is essential for GCF, LCM, and many other arithmetic problems. Memorize: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
What's the difference between GCF and LCM?
GCF (Greatest Common Factor) is the largest number that divides into both numbers evenly — used for simplifying fractions or finding common shared factors. LCM (Least Common Multiple) is the smallest number that both numbers divide into evenly — used for finding common denominators and certain scheduling problems. GCF(8,12) = 4; LCM(8,12) = 24.